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Exploding Julia sets in the Dynamics of fλ(z) = λJ1(iz)/iz M. Guru Prem Prasad? , Tarakanta Nayak† and Ashis Kumar Roy Department of Mathematics Indian Institute of Technology Guwahati, Guwahati 781 039 Abstract— In the present paper, we study the dynamics of the one parameter family of entire functions {fλ (z) = λf (z) : f (z) = J1 (iz)/iz for z ∈ C and λ is a non-zero real number} where J1 (z) is the Bessel function of the first kind of order one. We have found a critical parameter λ∗ ≈ 2.598 and show that the Julia set of fλ is a nowhere dense subset of the complex plane C for 0 < |λ| ≤ λ∗ and is equal to extended b = C ∪ {∞} for |λ| > λ∗ . This sudden change in the Julia complex plane C sets is known as explosion in the Julia sets or chaotic burst in the dynamics. Keywords—Complex dynamics, Julia set, Chaotic burst.

I. I NTRODUCTION A dynamical system is a physical setting together with rules for how the setting changes or evolves from one moment of time to the next or from one stage to the next. A basic goal of the mathematical theory of dynamical systems is to determine or characterize the long term behavior of the system. The simplest model of a dynamical process supposes that (n + 1)-th state, zn+1 can be determined solely from the knowledge of the previous state zn , that is zn+1 = f (zn ) where f is a function. These systems are called Discrete Dynamical Systems. We shall deal with one such systems, namely Complex Dynamical System where the the function f is a complex valued function of one complex variable. In the study of Complex Dynamical Systems, the evolution of the system is realized by the iterations of entire complex functions f : C → C. Entire functions are functions that are analytic b = C S{∞}, the sequence everywhere in C. For a point z0 ∈ C of iterates of z0 (or orbit of z0 ) is given by z0 = f 0 (z0 ), z1 = f (z0 ), z2 = f (z1 ) = f (f (z0 )) and zn = f (zn−1 ) = f n (z0 ) for n ≥ 3 where f n is the n-th iterate of f . The complex dynamics problem is to study the long term behavior of the sequence of b The set of all iterates of z0 for any given initial point z0 in C. b points in C whose sequences of iterates exhibit stable behavior b whose seis called the Fatou set and the set of all points in C quences of iterates exhibit unstable or chaotic behavior is called the Julia set. The following two definitions give a precise mathematical meaning to this idea. Definition I.1: A family T of analytic functions defined in a domain D ⊆ C is said to be normal at a point z0 ∈ D if every sequence extracted from T has a subsequence which converges uniformly either to a bounded function or to ∞ on each compact subset of some neighborhood of z0 . Definition I.2: The Fatou set of an entire function f (z), is denoted by F(f ), is defined as b : the sequence of iterates {f n } is normal at z} F(f ) = {z ∈ C The complement of the Fatou set F(f ) in the extended complex b is known as the Julia set of f and is denoted by J(f ). plane C ? Author

for correspondence. Tel:(361)2582608, Email: [email protected] research work of Tarakanta Nayak is supported by the CSIR Senior Research Fellowship No.9/731(31)/2004-EMR-I. † The

The point at ∞ is always in the Julia set since it is an essential singularity for which f can not be defined there. The Fatou set of a function is open by definition. The Julia set is always a non empty and perfect set. Also the interior of the ˆ [2]. Julia set is empty, unless it is whole of C The dynamics of a function is effectuated basically by the periodic points of the function. The definition and the nature of the periodic points are given below. Definition I.3: A point z is called a p-periodic point of f if p is the smallest natural number such that f p (z) = z. If p = 1, z is called a fixed point. A p-periodic point z is said to be attracting, indifferent or repelling if |(f p )0 (z)| < 1, = 1 or > 1 respectively. Further, an indifferent p-periodic point is called rationally (irrationally) indifferent if (f p )0 (z) = ei2πt where t is rational (irrational). A rationally indifferent periodic point is also called parabolic periodic point. A Fatou component is a maximal connected open subset of F(f ). A component U0 of F(f ) is p-periodic if p is the smallest natural number such that f p (U0 ) ⊆ U0 . The set {U0 , U1 = f (U0 ), U2 = f 2 (U0 ), · · · , Up−1 = f p−1 (U )} is called a pperiodic cycle ofT Fatou components. If U is a Fatou component such that f p (U ) f q (U ) = ∅ for all natural numbers p and q, then U is called a wandering domain. The classification of periodic Fatou components for transcendental entire functions is given below (See also: [2]). Suppose that U is a p-periodic Fatou component. Then exactly one of the following possibilities occur. 1. Attracting Basin: If for all points z in U , limn→∞ f np (z) = z ∗ where z ∗ is an attracting p-periodic point lying in U , then the component U is called an attracting basin. 2. Parabolic domain: In this case ∂U (the boundary of U ) contains a rationally indifferent p-periodic point z ∗ . Further limn→∞ f np (z) = z ∗ for all z ∈ U . 3. Baker Domain: If for all points z ∈ U , limn→∞ f np (z) = ∞ then the Fatou component U is called a Baker domain. 4. Rotational Domain: A Fatou component U is said to be a rotational domain if there exists an analytic homeomorphism φ : U → D such that φ(f p (φ−1 (z))) = ei2πα z for some irrational number α where D is either the unit disc or an annulus {z : 0 < r < |z| < 1}. In the first case, U is called Siegel disk and in the second case Herman ring. Entire functions do not have Herman rings [2]. Siegel disk is simply connected. Besides periodic points, the singular values and its forward orbits play an important role in determining the dynamics of a function. Definition I.4: A point z is a critical point of f if f 0 (z) = 0. The value of the function f at z, w = f (z) is called the critical value of f . A point w is called an asymptotic value of f if there exists a continuous curve γ(t) : (0, ∞) → C such that

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limt→∞ γ(t) = ∞ and limt→∞ f (γ(t)) = w. All the critical and asymptotic values of a function are known as singular values. The set of all singular values of a function f is denoted by Sf . The set of all forward orbits of all singular values is denoted by O+ (Sf ) and is given by {f n (w) : w ∈ Sf and n = 0, 1, 2, · · · }. The relation between the set O+ (Sf ) and the periodic Fatou components of f is summarized in the following theorem. Theorem I.1: [2] Let f be an entire function and C = {U0 , U1 , · · · , Up−1 } be a p-periodic cycle of components of F(f ). 1. If C is a cycle of attracting basins or parabolic domains, then there exists aTnatural number j with j ∈ {0, 1, · · · , p − 1} such that Uj Sf 6= ∅. 2. If C is a cycle of rotational domains then ∂Uj ⊂ O+ (Sf ) for all j ∈ {0, 1, · · · , p − 1}. In recent years, the dynamics of transcendental entire functions has been studied by many researchers. While studying the dynamics of one parameter family E ≡ {λez : λ > 0}, Devaney and coworkers [5] observed that the Julia set of λez is a nowhere dense subset of extended complex plane for 0 < λ < 1/e, where as it becomes the whole of the extended complex plane for λ > 1/e. This sudden change in the Julia sets is known as explosion in the Julia sets or chaotic burst in the dynamics of one parameter family E. Similar chaotic z bursts are exhibited for the family {fλ (z) = λ e z−1 : λ > 0} by Kapoor and Prasad [9] and for the family {fλ (z) = λ sinh (z) : λ is a non zero real parameter} by Prasad [7]. z

In the present paper, we study the dynamics of the one parameter family of entire functions {fλ (z) = λf (z) : f (z) = J1 (iz)/iz for z ∈ C and λ is a non-zero real number} where J1 (z) is the Bessel function of the first kind of order one given ¡ z ¢2k+1 k P∞ for z ∈ C. We remark that by J1 (z) = k=0 k!(−1) (k+1)! 2 (iz) f (z) = J1iz = z −1 I1 (z) where I1 (z) denotes the modified Bessel function of first kind and order one. Clearly ∞

f (z) =

J1 (iz) X z 2k = for z ∈ C iz 22k+1 k! (k + 1)! k=0

is an entire function. II. DYNAMICS OF fλ J1 (iz) for z ∈ C and Let B ≡ {fλ (z) = λf (z) : f (z) = iz λ is a non-zero real number}. For fλ ∈ B, observe that fλ (−z) = fλ (z). So, f−λ (z) = −fλ (z) = −fλ (−z) for all z ∈ C. Consen quently, f−λ (z) = −fλn (−z) for all z ∈ C and n ∈ N, and dynamics of fλ and f−λ are essentially same. The functions fλ and f−λ are called conformally conjugate. So, it is sufficient to study the dynamics of the one parameter family J1 (iz) for z ∈ C and B+ ≡ {fλ (z) = λf (z) : f (z) = iz λ > 0}. We first prove that the function fλ has infinitely many singular values in Proposition II.1. The existence and nature of the fixed points for fλ is proved in Theorem II.1. Non-existence of

certain type of periodic components in Fatou set of fλ is established in Propositions II.2 and II.3. Finally, a complete picture of the dynamics of the functions fλ is presented. A. Singular values of fλ The following proposition locates all singular values of fλ ∈ B+ . Proposition II.1: Let fλ ∈ B+ . Then, fλ has infinitely many singular values all lying in a bounded set of R. J2 (iz) Proof: We first observe that fλ0 (z) = −λi(iz) where J2 (z) is the Bessel function of the first kind of order two [4]. The critical points of fλ (z) are the solutions of J2 (iz) = 0 and these are infinitely many purely imaginary numbers [4]. They form an unbounded sequence as J2 (iz) is entire. Let these be arranged in an increasing sequence in magnitude, namely, {zk = ixk } where xk ∈ R for all k ∈ N. Now, the critical value corresponding to the critical point zk is given by 1 (−xk ) fλ (zk ) = fλ (ixk ) = λ J(−x which is a real number. Since k) limk→∞ fλ (zk ) = 0 [4] and fλ (zk ) 6= 0 for all k, there are infinitely many critical values of fλ . Since J1x(x) is bounded on R, all the critical values lie in an bounded interval in R. It is easy to show that the order (which measures the growth of maximum modulus) [8] of the entire function fλ (z) is one. By Ahlfors-Denjoy theorem [1], it follows that fλ has at most two finite asymptotic values. The function fλ tends to 0 when z tends to ∞ along the positive and the negative imaginary axis. So, 0 is an asymptotic value for fλ . If a 6= 0 is an asymptotic value of f , then −a and a ¯ will be also asymptotic values since fλ (z) = fλ (−z) and fλ (¯ z ) = fλ (z). This is not possible by Ahlfors-Denjoy theorem [1]. Therefore, fλ has only one finite asymptotic value, namely, 0. This completes the proof. B. Real Periodic Points of fλ In this subsection, the existence and nature of real periodic points of fλ is studied. The function f (x) = J1 (ix)/(ix) takes the positive values for all x ∈ R. It gives that, all the real periodic points of fλ (x) lie on the positive real axis. Suppose x0 is a real periodic point such that fλp (x0 ) = x0 for some p ≥ 1. Since fλ0 (x) > 0 for x > 0, fλp (x0 ) = x0 is not possible for p > 1. Therefore, any real periodic point of fλ is a fixed point. Consider the function φ(x) = f (x) − xf 0 (x) for x > 0. As φ0 (x) = −xf 00 (x) < 0 for all x > 0, φ(x) is decreasing for x > 0. Using the intermediate value theorem and the facts that φ(0) = f (0) > 0 and limx→∞ φ(x) = −∞, we get a unique point x∗ > 0 such that   > 0 for 0 ≤ x < x∗ = 0 for x = x∗ φ(x)  < 0 for x > x∗ 1 ∗ is Throughout this paper, we denote λ∗ by f 0 (x ∗ ) where x 0 the unique positive real root of φ(x) = f (x) − xf (x) = 0. Note that 0 < λ∗ < f 01(0) , since x∗ > 0 and f 01(x) is decreasing in R+ . Numerically it is found that λ∗ ≈ 2.598. The following theorem describes the existence and nature of the real fixed points of fλ for λ > 0. Theorem II.1: Let fλ (x) = λJ1 (ix)/(ix) for x ∈ R where λ > 0. Then,

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1. For 0 < λ < λ∗ , the function fλ has only two real fixed points aλ and rλ (say) with 0 < aλ < rλ where aλ is attracting and rλ is repelling. 2. For λ = λ∗ , the function fλ has only one real fixed point at x∗ , and it is rationally indifferent. 3. For λ > λ∗ , the function fλ has no real fixed point. Proof: Let gλ (x) = fλ (x)−x for x ∈ R. Since all the coefficients of the Taylor series of fλ about the point z = 0 are non-negative, the functions fλ (x), fλ0 (x) and fλ00 (x) are positive for x > 0. It gives that fλ (x) and fλ0 (x) are increasing in R+ , the positive real axis. Suppose that λ < f 01(0) . Then gλ0 (0) < 0. Since the function gλ0 (x) = fλ0 (x) − 1 is increasing in R+ and tends to +∞ as x approaches to +∞, there exists a unique xλ > 0 such that gλ0 (x) < 0 for x ∈ (0, xλ ), gλ0 (xλ ) = 0 and gλ0 (x) > 0 for x ∈ (xλ , ∞). Therefore, gλ decreases in the interval [0, xλ ], attains its minimum value at xλ and then increases in the interval (xλ , ∞). For each λ < f 01(0) there exists a unique positive real 1 number xλ such that λ = f 0 (x . λ) 1 1 ∗ 1. If λ < λ , then f 0 (xλ ) < f 0 (x Since f 01(x) is strictly ∗) . decreasing in R+ , it follows that xλ > x∗ and φ(xλ ) < φ(x∗ ) = λ) 0 as φ(x) is decreasing. Since f 0 (xλ ) > 0 and for that, fφ(x 0 (x ) = λ

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Fig. 2. Graph of fλ for λ = λ∗ ≈ 2.598 and the line y = x. 14

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Fig. 3. Graph of fλ for λ >

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and the line y = x, (λ = 2.8).

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C. Fatou Components of fλ

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0

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Fig. 1. Graph of fλ for λ <

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and the line y = x, (λ = 2).

gλ (xλ ) < 0, the minimum value gλ (xλ ) = fλ (xλ ) − xλ of gλ (x) is negative. Therefore, there exist two points aλ and rλ (say) with aλ < xλ < rλ such that gλ (aλ ) = 0 = gλ (rλ ). That means, the points aλ and rλ are the fixed points of fλ (x) 0 0 0 (See Fig. 1). Observe that fλ (aλ ) < fλ (xλ ) = 1 and fλ (xλ ) > 0 fλ (rλ ) = 1. Therefore, aλ is attracting and rλ is repelling. 2. By similar arguments as in proof of (1), we conclude that gλ (xλ ) = 0 for λ = λ∗ and xλ = x∗ . As gλ (xλ ) is the minimum value of gλ (x), xλ is the only zero of gλ (x). Hence fλ (x) has only one real fixed point x∗ (See Fig. 2) and it is rationally indifferent. 1 1 3. For λ∗ < λ < f 01(0) , f 0 (x ∗ ) < f 0 (x ) . It implies that xλ < λ ∗ x and, consequently, φ(xλ ) > 0. Further, gλ (x) > gλ (xλ ) = 0 for all x > 0. Therefore, there is no real fixed point of fλ (x) for f 01(0) > λ > λ∗ . For λ ≥ f 01(0) , gλ0 (0) ≥ 0 and gλ (x) > gλ (0) ≥ 0 for all x > 0 as gλ is increasing in positive real axis. So fλ has no fixed point in R (See Fig. 3).

We show in this subsection that the Fatou set of fλ does not contain certain kinds of Fatou components. Proposition II.2: Let fλ ∈ B+ . Then, the Fatou set F(fλ ) does not contain any Siegel disk. Proof: As λ > 0, fλ (x) > 0 for all x ∈ R. By Proposition II.1, the set of all singular values of fλ is contained in R+ . Consequently, the forward orbits of all singular values O+ (Sfλ ) is also contained in R+ . Let U be a Siegel disk in the Fatou set of fλ . Then fλ is a bijection on U by definition. It follows from Picard’s theorem that, there are infinitely many pre-periodic components in F(fλ ) each of which is equal to fλ−k (U ) for some k ∈ N. It is known from Theorem I.1 that O+ (Sfλ ) is dense in ∂U , the boundary of U . So ∂U is contained in R+ . But this is not possible since U is simply connected. Therefore, the Fatou set of fλ for λ > 0 does not contain a Siegel disk. Now, we prove non-existence of Baker domains and wandering domains for fλ ∈ B+ . Proposition II.3: Let fλ ∈ B+ . Then, F(fλ ) contains neither wandering domain nor Baker domain. Proof: Let on contrary, W be a wandering domain of fλ . It is already shown that any real number tends either to a non repelling (attracting or parabolic) real fixed point or to ∞ under iteration of fλ . It is known that for a wandering domain W of an entire function f , every limit function of {f n (z)}n>0 for z ∈ W is in Julia set and is a limit point of O+ (Sf ) (∞ can well be such a limit point) [3]. Let z ∈ W and {fλnk (z)}k>0

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converges to z0 . Then, z0 is a real number and in Julia set. Being a limit point of the set {fλn (z)}n>0 , z ∈ W ⊂ F(fλ ), z0 can not be a repelling fixed point. Therefore, z0 is not a fixed point of fλ and, hence limm→∞ f m (z0 ) = ∞ as fλ is increasing in the positive real axis. By continuity of fλ , it follows that limk→∞ f nk +m (z) = f m (z0 ). Since limk→∞ f nk +m (z) = f m (z0 ) and limm→∞ f m (z0 ) = ∞, we can find a subsequence {fλmk (z)}k>0 of {fλn (z)}n>0 such that limk→∞ fλmk (z) = ∞. Using logarithmic change of variable, it has been proved that, if f is an entire function and the set of all its singular values is bounded then the sequence of iterates {f n (z)}n>0 can not converge to ∞ for any z in the Fatou set of f [6]. From the proof of this result, we can show that the above result is true for any subsequence {fλmk (z)}k>0 . It contradicts the conclusion made in the previous paragraph. Therefore, fλ has no wandering domain. Non-existence of Baker domains of any period follows from the fact that no subsequence of {fλn (z)}n>0 can converge to ∞ for any z ∈ F(fλ ) [6]. D. Fatou and Julia sets of fλ In this subsection, the dynamics of fλ (z) for z ∈ C is described for each non zero real number λ. Theorem II.2: Let fλ ∈ B+ . Then, 1. For 0 < λ < λ∗ , the Fatou set F(fλ ) is equal to the attracting basin A(aλ ) of the real attracting fixed point aλ . 2. For λ = λ∗ , the Fatou set F(fλ ) is equal to the parabolic domain P (x∗ ) corresponding to the real rationally indifferent fixed point x∗ . 3. For λ > λ∗ , the Fatou set F(fλ ) is empty. Proof: The Fatou set of fλ does not contain any Siegel disk, Baker domain or wandering domain by Propositions II.2 and II.3. So, every periodic Fatou component is an attracting basin or parabolic domain. All the singular values of fλ and their forward orbits lie in the real axis. If there is a periodic attracting basin or parabolic domain U , say corresponding to a non-real periodic point then there is a singular value w of fλ in U by Theorem I.1. In that case fλn (w) must be non-real for sufficiently large n which is not possible. So, any periodic Fatou component corresponds only to a real attracting or a parabolic periodic point. Further, it is shown that any real periodic point of fλ is a fixed point. 1. By Theorem II.1(1), fλ has only two real fixed points namely aλ and rλ . Let A(aλ ) = {z ∈ C : fλn (z) → aλ as n → ∞} be the attracting basin of the real attracting fixed point aλ . Since there are no other attracting or parabolic real fixed points, F(fλ ) = A(aλ ). 2. By Theorem II.1(2), fλ has only one real rationally indifferent fixed point x∗ . Let P (x∗ ) = {z ∈ C : fλn (z) → x∗ as n → ∞} be the parabolic domain corresponding to the real rationally indifferent fixed point. Since there are no other attracting or parabolic real fixed points, F(fλ ) = P (x∗ ) for λ = λ∗ . 3. By Theorem II.1(3), fλ has no real fixed point. So F(fλ ) = ∅. The following result gives an algorithm to computationally generate the pictures of the Julia sets of fλ .

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Corollary II.1: Let fλ ∈ B+ . Then, 1. For 0 < λ < λ∗ , the Julia set J(fλ ) is equal to the complement of the attracting basin A(aλ ) of the real attracting fixed point aλ . 2. For λ = λ∗ , the Julia set J(fλ ) is equal to the complement of the parabolic basin P (x∗ ) of the real rationally indifferent fixed point x∗ . b 3. For λ > λ∗ , the Julia set J(fλ ) = C. The following theorem is an easy consequence of Theorem II.2 and the conformal conjugacy between fλ and f−λ . Theorem II.3: Let λ < 0. Then, 1. For −λ∗ < λ < 0, the Fatou set F(fλ ) is equal to the attracting basin A(−aλ ) of the real attracting fixed point −aλ . 2. For λ = −λ∗ , the Fatou set F(fλ ) is equal to the parabolic basin P (−x∗ ) of the real rationally indifferent fixed point −x∗ . 3. For λ < −λ∗ , the Fatou set F(fλ ) is empty. Corollary II.2: Let λ < 0. Then, 1. For −λ∗ < λ < 0, the Julia set J(fλ ) is equal to the complement of the attracting basin A(−aλ ) of the real attracting fixed point −aλ . 2. For λ = −λ∗ , the Julia set J(fλ ) is equal to the complement of the parabolic basin P (−x∗ ) of the real rationally indifferent fixed point −x∗ . b 3. For λ < −λ∗ , the Julia set J(fλ ) = C. III. C ONCLUSION In the previous section, it is proved that the Julia set J(fλ ) is equal to the complement of either the attracting basin or the parabolic domain for 0 < |λ| ≤ λ∗ . Since the Fatou set of fλ is non-empty for 0 < |λ| ≤ λ∗ , it follows that the Julia set of fλ has empty interior. That is, the Julia set of fλ is a nowhere dense subset of the complex plane for 0 < |λ| ≤ λ∗ . If |λ| crosses the value λ∗ , the Julia set suddenly explodes and equals to the extended complex plane. Thus, an explosion in the Julia sets or chaotic burst in the dynamics of the one parameter (iz) for z ∈ family B ≡ {fλ (z) = λf (z) : f (z) = J1iz C and λ is a non-zero real number} of entire transcendental functions occurs at |λ| = λ∗ . R EFERENCES ¨ ber die Asymptotischen Werte der Meromorphen Functio[1] L. V. Ahlfors, U nen Endlicher Ordnung, Acta. Acad. abo. Math. phys. 6(1932), 3-8. [2] W. Bergweiler, Iteration of Meromorphic Functions, Bulletin of American Mathematical Society 29(1993), 2, 151–188. [3] W. Bergweiler, Mako Haruta, Hartje Kriete, Hans-G¨ unter Meier and Nobert Terglane On the limit functions of Iterates in wandering domains, Annales Academiae Scientiarum Fennicae Mathematica Series A. I. Mathematica Volumen 18(1993), 369–375. [4] F. Bowman, Introduction to Bessel Functions, Dover, 1958. [5] R. L. Devaney and M. B. Durkin, The Exploding Exponential and Other Chaotic Bursts in Complex Dynamics, Amer. Math. Monthly 98(1991), 4, 217-233. [6] A. E. Eremenko and M. Yu. Lyubich, Dynamical Properties of Some Classes of Entire Functions, Ann. Inst. Fourier, Grenoble 12(1992), 4, 9891020. [7] M. Guru Prem Prasad, Chaotic Burst in the Dynamics of fλ (z) = sinh (z) λ z , Regular and Chaotic Dynamics 10(2005), 1, 71-80. [8] A. S. B. Holland, Introduction to The Theory of Entire Functions, Academic Press, 1973. [9] G. P. Kapoor and M. Guru Prem Prasad, Dynamics of (ez − 1)/z: the Julia sets and Bifurcation, Ergodic Theory and Dynamical Systems 18(1998), 1363-1383.